Macroscopic dimension and duality groups

Abstract

We show that for a rationally inessential orientable closed $n$-manifold $M$ whose fundamental group $\pi$ is a duality group the macroscopic dimension of its universal cover is strictly less than $n$:$$ \dim_{MC}\Wi M<n.$$ As a corollary we obtain the following 0.1 Theorem. The inequality $ \dim_{MC}\Wi M<n$ holds for the universal cover of a closed spin $n$-manifold $M$ with a positive scalar curvature metric if the fundamental group $\pi_1(M)$ is a virtual duality group virtually satisfying the Analytic Novikov Conjecture.